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Capital Structure in Banks 189 c = (1 – α)-quantile of the standard normal distribution 246 V = Value of the portfolio and which is also depicted for confidence level α 1 in Figure 5.8. Since we are looking at a one-day period in this section, we have ignored time index H in Equation (5.28). For the same reason, the expected return µ will be very small 247 and is, therefore, almost always ignored when calculating the daily VaR (hence VaR reduces to c⋅σ R ⋅V). 248 Assuming that the returns of the positions constituting the portfolio are distributed normally greatly reduces the bottom-up calculation of the overall portfolio VaR P : VaR = ∑∑VaR VaR ρ = VaR T ⋅Σ ⋅VaR P i j ij i j (5.29) where ρ ij = Correlation between value changes in assets i and j Σ = Correlation matrix of value changes in the portfolio positions VaR = Vector of single transaction VaR (T indicates the transposed vector thereof) This approach is called the parametric approach 249 and clearly shows that only the marginal contribution (expressed by the correlation with the remainder of the portfolio 250 ) of a single position to the overall portfolio risk counts. 251 However, the normality assumption is just an approximation of the true 246 c is, for example, for α = 2.5% roughly 1.96, for α = 1% roughly 2.33, and for α = 0.03% roughly 3.43, as can be read off any statistics table of the standard normal distribution; see, for example, Bamberg and Baur (1991), p. 313. 247 Compared to the volatility; see Stulz (2000), p. 4-13. 248 See, for example, Hirschbeck (1998), p. 154, and Dowd (1998), pp. 41 and 43. 249 See, for example, Dowd (1998), pp. 42 and 63–98. For a discussion of the various variants such as the delta-normal or the delta-gamma approach to include nonlinear positions see Wilson (1996), pp. 193–232. 250 Another way to calculate the marginal contribution of a single position to the overall portfolio risk is to calculate VaR for the portfolio without the position, next for the portfolio including the position, and then build the difference between the two. However, this can be cumbersome and difficult to do; see Dowd (1998), pp. 48–50, for a discussion and the derivation of a shortcut formula in which the change in VaR = marginal VaR = w A β A,P VaR without , where w A = value weight of new asset A with respect to the portfolio value, β A,P = σ A,P /σ 2 P , and VaRwithout = portfolio VaR without new asset A. 251 Hirschbeck (1998), p. 156, also defines this equation for a combination of long and short positions.
190 RISK MANAGEMENT AND VALUE CREATION IN FINANCIAL INSTITUTIONS (empirical) distribution, which usually has fatter tails than assumed by normality, that is, losses exceeding VaR happen more often than predicted by the normal distribution. Similarly, nonlinear positions, such as options, cannot be easily included in the parametric approach. Therefore, banks also apply so-called Historical 252 and Monte Carlo simulations 253 to include both the nonlinearity of the positions and deviations from normality. 254 It is not the purpose of this section to discuss and evaluate these various approaches 255 to calculating portfolio VaR for market risk in detail. However, all three approaches use historical time series 256 and a number of assumptions (e.g., that correlations are stable over time) to estimate the input parameters. They are, therefore, subject to misspecification and estimation errors. Potential remedies are to conduct either all three “alternative” VaR approaches, 257 or both backtesting 258 and stress-testing 259 to validate that the model works appropriately. 260 Typically, the confidence interval (1 – α) chosen for the internal management and limit purposes of market risk is the 97.5% or 99% level. Even though this choice seems somewhat arbitrary, it is not the level that is important, but rather that it is applied consistently across time and products. At these levels, we can expect that VaR can and will be exceeded on an average of 5 to 6 days per year for the 97.5% confidence level and 2 to 3 days per annum for the 99% confidence level. Even though the estimates at these confidence levels are much more reliable than at higher confidence 252 See Hirschbeck (1998), pp. 182–187, Dowd (1998), pp. 99–107. 253 See Dowd (1998), pp. 108–120, Hirschbeck (1998), pp. 177–182. 254 Even though the Monte Carlo simulation can include non-normal positions, it generates random draws on the basis of a correlation matrix that assumes normality and does therefore not model deviations from normality. 255 For such an evaluation See for example, Hirschbeck (1998), pp. 188–196. 256 Even the Monte Carlo Simulation uses correlation and distribution assumptions that are typically based on historical time series. 257 The three results should be compared to find out what the “real” VaR is. Some banks take the most conservative estimate out of the three results. 258 This procedure evaluates whether the number of losses greater than VaR are in line with the prediction by the assumed probability distribution. It can therefore identify whether the underlying model is correct. For a detailed discussion of these methods see Stahl and Traber (2000), pp. 85–106. 259 For a stress test, a standard change in the risk factors is set (e.g., ± 5%, ±10%) and the change in portfolio value is calculated. There are two drawbacks: (1) this procedure does not relate the movements to a probability of occurrence and (2) it only assumes a simple (0; 1) correlation with other moves. For a discussion see for example, Dowd (1998), pp. 121–138, or Hirschbeck (1998), pp. 198–200. 260 If the models do not work appropriately, they need to be adjusted accordingly.
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isk management and value creation i
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Copyright © 2002 by John Wiley & S
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preface From an empirical as well a
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acknowledgments No book is solely t
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x CONTENTS Goals of Risk Management
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xii CONTENTS Definition of RAROC 24
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xiv FIGURES Figure 5.5 Figure 5.6 F
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symbols ↑ ↓ α β i xvi Increas
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xviii SYMBOLS ln Natural logarithm
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abbreviations APT BIS bn bps CAPM C
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2 RISK MANAGEMENT AND VALUE CREATIO
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CHAPTER 2 Foundations for Determini
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Foundations for Determining the Lin
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eferences Aguais, Scott D., and Ant
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References 295 Blanden, Michael (19
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References 297 and Abuses,” Journ
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References 299 Froot, Kenneth A., a
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References 301 risiken aus Handelsg
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References 303 Investments in Stock
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References 305 Institute of Technol
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References 307 Sharpe, William F. (
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References 309 Wall, L.D., and John
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312 INDEX Approach (continued) top-
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314 INDEX Capital: Tier-1, 139, 146
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316 INDEX Credit: derivatives, 178,
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318 INDEX Expectations: homogeneous
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320 INDEX Incentive(s), 22, 33, 45,
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322 INDEX Managers, poorly diversif
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324 INDEX Overinvestment, 75, 79, 8
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326 INDEX Reserve(s) (continued) lo
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328 INDEX Risk, liquidity, 3, 165 R
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330 INDEX Sharpe ratio, 244, 252 Sh
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332 INDEX Value at risk (VaR) (cont
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